login
A308411
Number of ways to write n as 2^i*3^j + A008347(k), where i, j and k > 0 are nonnegative integers.
6
0, 1, 2, 3, 4, 4, 4, 4, 5, 5, 5, 3, 5, 5, 4, 5, 5, 3, 6, 7, 6, 6, 5, 6, 7, 4, 5, 8, 6, 8, 7, 5, 5, 8, 8, 6, 7, 5, 7, 8, 7, 6, 6, 6, 8, 6, 5, 8, 6, 7, 7, 5, 8, 7, 8, 10, 9, 9, 6, 7, 8, 8, 6, 7, 10, 8, 8, 6, 9, 6, 7, 9, 8, 7, 7, 8, 9, 4, 8, 11, 10, 6, 9, 7, 11, 8, 8, 10, 9, 6, 4, 7, 10, 7, 7, 3, 11, 9, 10, 7
OFFSET
1,3
COMMENTS
Conjecture: a(n) > 0 for all n > 1.
We have verified this for all n = 2..10^6.
Qing-Hu Hou at Tianjin University extended the verification to 2*10^7. Then I used Hou's program to verify a(n) > 0 for n up to 10^9. - Zhi-Wei Sun, May 28 2019
Conjecture verified for n up to 10^10. - Giovanni Resta, May 28 2019
LINKS
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.
EXAMPLE
a(2) = 1 with 2 = 2^0*3^0 + A008347(2).
a(3) = 2 with 3 = 2^0*3^0 + A008347(1) = 2^1*3^0 + A008347(2).
MATHEMATICA
f[n_]:=f[n]=FactorInteger[n];
FQ[n_]:=FQ[n]=n>0&&Part[f[n], Length[f[n]]][[1]]<4;
s[0]=0; s[n_]:=s[n]=Prime[n]-s[n-1];
tab={}; Do[r=0; Do[If[FQ[n-s[k]], r=r+1], {k, 1, 2n-1}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 25 2019
STATUS
approved